--- title: "Three triaxial tests, Mohr circles, φ" subject: "Soil Mechanics" chapter: 10 key: "semester_4/soil_mechanics:mohr-circle-three-triaxial-tests-friction" source: https://www.bctengineer.com/q/semester_4/soil_mechanics/mohr-circle-three-triaxial-tests-friction site: BCT Engineering --- # Three triaxial tests, Mohr circles, φ Soil Mechanics · Chapter 10: Shear Strength of Soil - Asked in: 2079 Asoj, 8 marks - Video: [Mohr's Circle for Triaxial test](https://www.youtube.com/watch?v=Av0bsVLarZk) — Curious Civil Engineer - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/mohr-circle-three-triaxial-tests-friction - Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-10 ## Question A series of shear tests was performed on a soil. Each test was carried out until the soil sample sheared and the stresses for each test are as follows. | Test | Cell pressure $\sigma_3$ (kN/m²) | Deviator stress (kN/m²) | |---|---|---| | 1 | 300 | 875 | | 2 | 400 | 1160 | | 3 | 500 | 1460 | Plot the Mohr circle of stress and the strength envelope and determine the angle of internal friction of the soil. ## Short answer For each test, σ1 = σ3 + deviator stress: 1175, 1560, 1960 kN/m². Using sinφ = (σ1-σ3)/(σ1+σ3) gives φ of 36.4°, 36.3°, 36.4°. The strength envelope passes near origin, so c≈0, φ≈36.4°. ## Answer **Step 1: Principal stresses at failure** ($\sigma_1 = \sigma_3 + \sigma_d$) | Test | $\sigma_3$ (kN/m²) | $\sigma_d$ (kN/m²) | $\sigma_1$ (kN/m²) | Centre $\frac{\sigma_1+\sigma_3}{2}$ | Radius $\frac{\sigma_d}{2}$ | |---|---|---|---|---|---| | 1 | 300 | 875 | 1175 | 737.5 | 437.5 | | 2 | 400 | 1160 | 1560 | 980 | 580 | | 3 | 500 | 1460 | 1960 | 1230 | 730 | **Step 2: Mohr circles and envelope.** Draw each circle with its centre on the $\sigma$-axis and radius as in the table, then draw the common tangent. ```text tau | ___..--''''-. | ,-'' ___...--'' `. | ,-' _.--'' .--'' . \ | ,' .-' __,-' _.-' \ \ | / .'.-' ,-' _.-' phi | | +--+---------------------------+-- sigma 0 300 400 500 1175 1560 1960 ``` The tangent passes (almost) through the origin, so $c \approx 0$ (a cohesionless soil). **Step 3: Angle of internal friction** $$ \sin\phi = \frac{\sigma_1-\sigma_3}{\sigma_1+\sigma_3} $$ $$ \begin{aligned} \text{Test 1: } \sin\phi &= \frac{875}{1475} = 0.593 \;\Rightarrow\; \phi = 36.4^\circ \\ \text{Test 2: } \sin\phi &= \frac{1160}{1960} = 0.592 \;\Rightarrow\; \phi = 36.3^\circ \\ \text{Test 3: } \sin\phi &= \frac{1460}{2460} = 0.594 \;\Rightarrow\; \phi = 36.4^\circ \end{aligned} $$ **Answer: $c \approx 0$ and $\phi \approx 36.4^\circ$ (about $36^\circ$).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the major principal stress at failure found from the data? A. sigma_1 = sigma_3 + deviator stress B. sigma_1 = sigma_3 - deviator stress C. sigma_1 = deviator stress / 2 D. sigma_1 = sigma_3 x deviator stress Answer: A. Deviator stress is sigma_1 - sigma_3, so test 1 gives 300 + 875 = 1175 kN/m². (Key point: Sigma_1 computation) 2. For test 1, what are the centre and radius of the Mohr circle? A. Centre 875 and radius 300 B. Centre 1175 and radius 437.5 C. Centre 300 and radius 875 D. Centre 737.5 and radius 437.5 kN/m² Answer: D. Centre = (1175 + 300)/2 = 737.5 and radius = 875/2 = 437.5. (Key point: Circle geometry) 3. What does the common tangent passing through the origin indicate? A. c is large B. phi = 0 C. c is about 0, a cohesionless soil D. The soil is saturated clay Answer: C. An envelope through the origin has no intercept, so c = 0. (Key point: Cohesion) 4. Which formula gives phi for each test, and what is the result for test 1? A. sin phi = (sigma_1 + sigma_3)/(sigma_1 - sigma_3) so phi = 90° B. sin phi = (sigma_1 - sigma_3)/(sigma_1 + sigma_3) = 875/1475 so phi = 36.4° C. tan phi = 875/1475 so phi = 30.7° D. sin phi = 875/1175 so phi = 48° Answer: B. For a circle tangent to an envelope through the origin, sin phi = radius/centre = 437.5/737.5 = 0.593, phi about 36.4°. 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